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Thread: Procedure to Get Derivative of exp(x) Respect to x

  1. #1 Procedure to Get Derivative of exp(x) Respect to x 
    Forum Freshman
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    Excuse me ... .

    The definition of derivative a function respect to is

    ... .

    Consider ... then

    ... .

    In real calculus, we have known that ... .

    So, we have to say that

    ... (*) ... .

    The problem is how to prove the last equality (*) ... .

    Maybe, we can prove the last equality by L'Hopital Theorem ... .

    But, the L' Hopital Theorem uses the concept of derivative ... .

    Whereas, the concept of derivative uses the concept of limit ... .

    How can the last equality (*) be proven without L'Hopital Theorem ... ?

    Is the last equation (*) an assumption ... ?

    Thank you very much ... .


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  3. #2  
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    The demonstration depends on the definition that you accept for exp(x) :
    • my favorite definition is that exp(x) is a function such as it is equal to its derivative. .
    • You can also define exp(x) as the inverse function of Log. The demo is easy using formula of the derivative of inverse function :
      . Check the conditions required before using this formula.


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  4. #3  
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    You could use the power series for e^x.
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  5. #4  
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    Excuse me ... .

    I have known that the definition of Euler number is ... .

    If we use the power series procedure, we must use concept of derivative ... because the in Maclaurin series exist derivative expression, that is

    ... .

    Using Maclaurin series, we have to seek derivative of ... , whereas we will prove that derivative of is ... .

    I'm sorry if I do mistakes ... . Thank you for the attention ... .
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  6. #5  
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    As suggested by mathman, another possible definition for exp(x) is the limit of the series
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